A problem for convex body in the Euclidean plane

Let us consider, in the Euclidean plane $\mathbf{E}_2$ a fixed convex body $\mathbf{K}_0$ and a system $ {\mathbf{K}_1,..., \mathbf{K}_m\}$ of $n$-dimensional convex bodies. Assume that the sets $ \mathbf{K}_i$ ($i=1,...,m$) have random positions, being stochastically independent and uniformly distributed in a limited domain of $\mathbf{E}_2$, and denote by $\mathcal{S}_m$ the area of the convex body $\mathcal{K}_{m}= \mathbf{K}_0\cap (\mathbf{K}_1 \cap \mathbf{K}_2\cap...\cap \mathbf{K}_m)$. The aim of this paper is the study of the the random variable $\mathcal{S}_m$.